If that is true, then they shouldn't need attuning. Or do you mean you are giving out DMG items that do need attuning but that you think of as not being very powerful?
If the latter case, the power level of your campaign is likely greater than envisioned. The fix would be to lift the limit on allowable attuned items as others have suggested, to match the power level of your campaign. A strategy for trialing rules is to add them to an item e.g. an item that requires attunement but lets the bearer attune two further items. If it proves to be a problem you can easily remove it. Or if it works, you can at some point pull it into your world physics.
If the former case, then it resolves itself. Minor items don't need attuning.
Maybe you haven't carefully read the DMG.
It's not that tough to find items in the DMG that require attunement that are not really that powerful. In our group, we have:
Headband of Intellect (uncommon) which gave the Ranger / Wizard +1 to hit with a few spells, at level 6 when he made level 4 wizard, he bumped his Int to 18 so that he could give the headband to the Rogue and use 3 other attuned items (he has the most in the group). Considering that the vast majority of +1 items in the group do not even require attunement (like +1 weapons), this is not a powerful item. It will become more powerful to the Rogue since it boosts his Int more, but it only has one spell that it affects.
Brooch of Shielding (uncommon). This has yet to affect the campaign at all.
Sword of Vengeance (uncommon), this is actually a cursed item.
Sword of Life Stealing (rare), the Cleric has yet to actually roll a 20 with it after most of a level. Since it is not a +1 weapon, it did not give +1 to hit and +1 damage. So although this is a rare item that requires attunement, I do not consider it powerful (considering that its power has never ever come into play, ever).
Bracers of Defense (rare). Our version is the +3 one from the Hoard of the Dragon Queen supplement, but they do not stack with anything except Dex (as written in HotDQ, the item was too powerful for some classes like Monks, Barbarians, Sorcerers, and Wizards).
I took over as DM at level 5 (and the PCs recently just got to level 6), but we also had a few previous DM crafted items (before the DMG came out):
Boots of Leaping and Landing (uncommon) similar to Boots of Striding and Springing. They were used to leap once and land once.
An elemental Ring that can be used for 3 encounters per day that turns certain types of damage resistance into damage vulnerability for ammunition weapons. It has been applicable twice in the campaign so far and has been around for 5 levels.
So yeah, I think that there are quite a few attunement items in the DMG (and also in our game from the previous DM) that probably shouldn't be attunement items. As a DM, I could go through whichever ones I hand out and say "this does not require attunement" even though the DMG says that it does, but that's too much effort both in trying to determine the utility of every attunement item, and in keeping notes on which ones I modified and which ones I did not.
The game designers thought that a +1 through +3 weapons should not require attunement, but these are items that increase damage 60+% of the time and to a considerable amount in that 5% to 15% chance where the weapon would not have hit, and also against monsters with non-magic weapon type resistances. Personally, I think that +1 to +3 weapons, although rather boring, are very powerful items because they improve the power of the PC most of the time. The two weapon fighter can use two +3 weapons and be very powerful because of it, and still have 3 other attuned items. Other class concepts do not necessarily have this advantage.
